Catalog of rank-4 temperaments: Difference between revisions

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Line 470:

[[Badness]] (Sintel): 0.454

~~=== Commas 540/539, 847/845 ===~~

~~[[Subgroup]]: 2.3.5.7.11.13~~

~~[[Comma list]]: 540/539, 847/845~~

~~{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 0 0 3 3 | 0 0 1 1 -1 -1 | 0 0 0 2 -4 -3 }}~~

~~: Mapping generators: ~2, ~3, ~5, ~13/11~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.8339{{c}}, ~3/2 = 701.4961{{c}}, ~5/4 = 386.5624{{c}}, ~13/11 = 291.6840{{c}}

~~: [[error map]]: {{val| -0.166 -0.625 -0.084 +0.772 -0.627 +1.848 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4439{{c}}, ~5/4 = 386.4201{{c}}, ~13/11 = 291.7287{{c}}

~~: error map: {{val| 0.000 -0.511 +0.106 +1.052 -0.321 +2.198 }}~~

~~{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}~~

~~[[Badness]] (Sintel): 1.04~~

=== Commas 540/539, 625/624 ===

Line 523:

Line 505:

[[Badness]] (Sintel): 0.802

=== Commas 540/539, 847/845 ===

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 540/539, 847/845

{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 0 0 3 3 | 0 0 1 1 -1 -1 | 0 0 0 2 -4 -3 }}

: Mapping generators: ~2, ~3, ~5, ~13/11

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8339{{c}}, ~3/2 = 701.4961{{c}}, ~5/4 = 386.5624{{c}}, ~13/11 = 291.6840{{c}}

: [[error map]]: {{val| -0.166 -0.625 -0.084 +0.772 -0.627 +1.848 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4439{{c}}, ~5/4 = 386.4201{{c}}, ~13/11 = 291.7287{{c}}

: error map: {{val| 0.000 -0.511 +0.106 +1.052 -0.321 +2.198 }}

{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}

[[Badness]] (Sintel): 1.04

== Pentacircle (896/891) ==

Line 578:

[[Badness]] (Sintel): 1.04

== Moctdelismic (1375/1372) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 1375/1372

{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 0 | 0 0 1 0 -3 | 0 0 0 1 3 }}

: mapping generators: ~2, ~3, ~5, ~7

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0558{{c}}, ~3/2 = 701.8991{{c}}, ~5/4 = 385.7504{{c}}, ~7/4 = 969.3743{{c}}

: [[error map]]: {{val| +0.056 -0.000 -0.452 +0.660 -0.334 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9100{{c}}, ~5/4 = 385.7894{{c}}, ~7/4 = 969.4163{{c}}

: error map: {{val| 0.000 -0.045 -0.524 +0.590 -0.437 }}

{{Optimal ET sequence|legend=1| 12, 19e, 27, 29, 31, 41, 60e, 68, 72, 140, 152, 224, 364, 436, 547, 588, 619, 843, 1055ce }}

[[Badness]] (Sintel): 0.723

== Lehmerismic (3025/3024) ==

Line 613:

Line 631:

[[Badness]] (Sintel): 0.272

== Vishdelismic (5632/5625) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 5632/5625

{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 0 | 0 0 1 0 -3 | 0 0 0 1 3 }}

: mapping generators: ~2, ~3, ~5, ~7

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8976{{c}}, ~3/2 = 702.1145{{c}}, ~5/4 = 386.7638{{c}}, ~7/4 = 969.0306{{c}}

: [[error map]]: {{val| -0.102 +0.057 +0.245 -0.000 -0.136 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1309{{c}}, ~5/4 = 386.7770{{c}}, ~7/4 = 969.0087{{c}}

: error map: {{val| 0.000 +0.176 +0.463 +0.183 +0.052 }}

{{Optimal ET sequence|legend=1| 12, 19e, 22, 31, 53, 84e, 87, 99e, 118, 130, 152, 239, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}

[[Badness]] (Sintel): 0.804

== Trimitone (8019/8000) ==

Line 638:

Line 674:

{{Mapping|legend=1| 1 0 0 0 6 -3 | 0 1 0 0 -6 6 | 0 0 1 0 3 0 | 0 0 0 1 0 -1 }}

~~: Mapping generators: ~2, ~3, ~5, ~7~~

[[Optimal tuning]]s:

Line 649:

Line 684:

[[Badness]] (Sintel): 0.875

=== Commas 351/350, 2080/2079 ===

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 351/350, 2080/2079

{{Mapping|legend=1| 1 0 0 0 6 1 | 0 1 0 0 -6 -3 | 0 0 1 0 3 2 | 0 0 0 1 0 1 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9538{{c}}, ~3/2 = 701.5852{{c}}, ~5/4 = 387.1356{{c}}, ~7/4 = 969.7495{{c}}

: [[error map]]: {{val| -0.046 -0.416 +0.730 +0.831 +0.301 -1.447 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.5936{{c}}, ~5/4 = 387.0975{{c}}, ~7/4 = 969.7451{{c}}

: error map: {{val| 0.000 -0.361 +0.784 +0.919 +0.413 -1.369 }}

{{Optimal ET sequence|legend=1| 19, 26, 39df, 46, 53, 58, 72, 111, 130, 183, 313 }}

[[Badness]] (Sintel): 0.677

== Kalismic (9801/9800) ==

Line 668:

Line 720:

[[Badness]] (Sintel): 0.0415

=== ~~Commas~~ 1716/1715, 2080/2079 ~~===~~

=== Briggs ===

Briggs tempers out [[1716/1715]] and [[2080/2079]], and is the join of [[pessoal]] and [[greenland]], among many others.

[[Subgroup]]: 2.3.5.7.11.13

Line 691:

Line 745:

[[Badness]] (Sintel): 0.256

=== Kalimera ===

Kalimera is a [[weak extension]] of kalismic by adding the [[harmonisma]], and also an extremely high accuracy detemperament of [[donar]] achieved by observing the [[ragisma]].

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 9801/9800, 10648/10647

{{Mapping|legend=1| 2 0 1 0 4 9 | 0 1 0 0 -2 -4 | 0 0 2 0 2 3 | 0 0 0 1 1 1 }}

: mapping generators: ~99/70, ~3, ~1053/560, ~7

[[Optimal tuning]]:

* [[WE]]: ~99/70 = 600.0006{{c}}, ~3/2 = 701.9470{{c}}, ~1053/560 = 1093.1617{{c}}, ~7/4 = 968.8431{{c}}

: [[error map]]: {{val| +0.001 -0.007 +0.010 +0.020 -0.043 +0.015 }}

* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9469{{c}}, ~1053/560 = 1093.1614{{c}}, ~7/4 = 968.8438{{c}}

: error map: {{val| 0.000 -0.008 +0.009 +0.018 -0.045 +0.013 }}

{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 104c, 132de, 144, 166, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 5144, 6380, 7828, 9334, 12018 }}

[[Badness]] (Sintel): 0.164

== Unisquary (12005/11979) ==

Line 697:

Line 771:

[[Comma list]]: [[12005/11979]]

{{Mapping|legend=1| 1 0 0 ~~0 0~~ | 0 1 0 2 2 | 0 0 1 ~~-1 -1~~ | 0 0 0 3 4 }}

{{Mapping|legend=1| 1 0 0 -6 -8 | 0 1 0 2 2 | 0 0 1 2 3 | 0 0 0 3 4 }}

: Mapping generators: ~2, ~3, ~5, ~11/7

: Mapping generators: ~2, ~3, ~5, ~44/35

[[Optimal tuning]]s:

* [[~~CTE~~]] ~~and~~ [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~031~~{{c}}, ~5/4 = 386.~~233~~{{c}}, ~11/7 = ~~783~~.~~508~~{{c}}

* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0306{{c}}, ~5/4 = 386.2325{{c}}, ~44/35 = 397.2749{{c}}

: [[error map]]: {{val| -0.000 +0.076 -0.081 +0.243 -0.475 +0.541 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0306{{c}}, ~5/4 = 386.2325{{c}}, ~44/35 = 397.2750{{c}}

: error map: {{val| 0.000 +0.076 -0.081 -0.475 +0.541 }}

{{Optimal ET sequence|legend=1| 12, 21, 23de, 26, 37, 46, 58, 72, 118, 130, 190, 239, 311, 441, 559, 752e, 870 }}

[[Badness]] (Sintel): 2.43

=== Hensquary ===

Line 710:

Line 789:

[[Comma list]]: 364/363, 1716/1715

{{Mapping|legend=1| 1 0 0 ~~0 0~~ -2 | 0 1 0 2 2 3 | 0 0 1 ~~-1 -1 -1~~ | 0 0 0 3 4 5 }}

{{Mapping|legend=1| 1 0 0 -6 -8 -12 | 0 1 0 2 2 3 | 0 0 1 2 3 4 | 0 0 0 3 4 5 }}

~~: Mapping generators: ~2, ~3, ~5, ~11/7~~

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~{{c}}, ~3/2 = 702.~~072~~{{c}}, ~5/4 = 386.~~528~~{{c}}, ~11/7 = ~~783~~.~~823~~{{c}}

* [[WE]]: ~2 = 1199.7826{{c}}, ~3/2 = 702.2831{{c}}, ~5/4 = 386.9208{{c}}, ~44/35 = 396.8581{{c}}

* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~253~~{{c}}, ~5/4 = 386.~~776~~{{c}}, ~11/7 = ~~783~~.~~875~~{{c}}

: [[error map]]: {{val| -0.217 +0.111 +0.172 +0.156 +1.443 -1.487 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2535{{c}}, ~5/4 = 386.7762{{c}}, ~44/35 = 397.0993{{c}}

: error map: {{val| 0.000 +0.298 +0.463 +0.531 +1.915 -1.166 }}

{{Optimal ET sequence|legend=1| 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369 }}

[[Badness]] (Sintel): 0.988

=== Ekasquary ===

Line 724:

Line 806:

[[Comma list]]: 1575/1573, 4459/4455

{{Mapping|legend=1| 1 0 0 ~~0 0 0~~ | 0 1 0 2 2 0 | 0 0 1 -~~1 -1 3~~ | 0 0 0 3 4 -5 }}

{{Mapping|legend=1| 1 0 0 -6 -8 10 | 0 1 0 2 2 0 | 0 0 1 2 3 -2 | 0 0 0 3 4 -5 }}

~~: Mapping generators: ~2, ~3, ~5, ~11/7~~

[[Optimal tuning]]s:

* [[~~CTE~~]] ~~and~~ [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 701.~~952~~{{c}}, ~5/4 = 386.~~139~~{{c}}, ~11/7 = ~~783~~.~~548~~{{c}}

* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.9524{{c}}, ~5/4 = 386.1388{{c}}, ~44/35 = 397.4088{{c}}

: [[error map]]: {{val| -0.000 -0.003 -0.175 -0.417 +0.639 +0.150 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9524{{c}}, ~5/4 = 386.1388{{c}}, ~44/35 = 397.4089{{c}}

: error map: {{val| 0.000 -0.003 -0.175 -0.417 +0.639 +0.150 }}

{{Optimal ET sequence|legend=1| 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072 }}

== ~~Semicanousmic~~ (14641/14580) ==

[[Badness]] (Sintel): 1.12

== Semicathartismic (14641/14580) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: [[14641/14580]]

{{Mapping|legend=1| 1 0 2 0 1 | 0 1 2 0 2 | 0 0 -4 0 -1 | 0 0 0 1 0 }}

{{Mapping|legend=1| 1 0 -2 0 0 | 0 1 2 0 2 | 0 0 4 0 1 | 0 0 0 1 0 }}

: Mapping generators: ~2, ~3, ~18/11, ~7

: Mapping generators: ~2, ~3, ~11/9, ~7

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~334~~{{c}}, ~18/11 = ~~854~~.~~555~~{{c}}, ~7/4 = 968.~~826~~{{c}}

* [[WE]]: ~2 = 1200.0495{{c}}, ~3/2 = 702.2793{{c}}, ~11/9 = 345.4721{{c}}, ~7/4 = 968.7265{{c}}

* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~292~~{{c}}, ~18/11 = ~~854~~.~~548~~{{c}}, ~7/4 = 968.~~756~~{{c}}

: [[error map]]: {{val| +0.049 +0.374 +0.133 -0.000 -1.188 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2918{{c}}, ~11/9 = 345.4516{{c}}, ~7/4 = 968.7556{{c}}

: error map: {{val| 0.000 +0.337 +0.077 -0.070 -1.283 }}

{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}

[[Badness]] (~~Smith~~): 0.~~351 × 10-6~~

[[Badness]] (Sintel): 6.40

==~~= Tridecimal semicanousmic =~~==

== Semiporwellismic (16384/16335) ==

Subgroup: 2.3.5.7.11~~.13~~

[[Subgroup]]: 2.3.5.7.11

Comma list: ~~2080~~/~~2079, 14641/14580~~

[[Comma list]]: 16384/16335

~~Mapping:~~ {{~~mapping~~| 1 0 2 0 ~~1 -6~~ | 0 1 2 0 2 3 | 0 0 -4 0 -1 3 | 0 0 0 1 0 1 }}

{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 0 -2 | 0 0 2 0 -1 | 0 0 0 1 0 }}

: Mapping generators: ~2, ~3, ~128/99, ~7

Optimal ~~tunings~~:

[[Optimal tuning]]s:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~{{c}}, ~3/2 = 702.~~469~~{{c}}, ~18/11 = ~~854~~.~~644~~{{c}}, ~7/4 = ~~968~~.~~959~~{{c}}

* [[WE]]: ~2 = 1199.7329{{c}}, ~3/2 = 702.3657{{c}}, ~128/99 = 442.1589{{c}}, ~7/4 = 969.3599{{c}}

* CWE: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~483~~{{c}}, ~18/11 = ~~854~~.~~642~~{{c}}, ~7/4 = ~~968~~.~~989~~{{c}}

: [[error map]]: {{val| -0.267 +0.144 +0.103 -0.000 +0.456 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5107{{c}}, ~128/99 = 442.2243{{c}}, ~7/4 = 969.4120{{c}}

: error map: {{val| 0.000 +0.556 +0.646 +0.586 +1.436 }}

{{Optimal ET sequence|legend=0| 87, 111, ~~181~~, ~~198~~, ~~323~~, ~~410~~ }}

{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}

Badness (~~Smith~~): 17.~~1 × 10-6~~

[[Badness]] (Sintel): 4.00

===~~= 17-limit =~~===

=== Tridecimal semiporwellismic ===

Subgroup: 2.3.5.7.11.13~~.17~~

[[Subgroup]]: 2.3.5.7.11.13

Comma list: ~~715~~/~~714~~, ~~1089~~/~~1088, 14641/14580~~

[[Comma list]]: 2080/2079, 3584/3575

~~Mapping:~~ {{~~mapping~~| 1 0 2 0 1 -~~6 -4 | 0 1~~ 2 0 ~~2 3 6~~ | 0 0 -4 0 -1 3 -2 | 0 0 0 1 0 1 0 }}

{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 1 0 -2 0 | 0 0 2 0 -1 -3 | 0 0 0 1 0 1 }}

Optimal ~~tunings~~:

[[Optimal tuning]]s:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~{{c}}, ~3/2 = 702.~~413~~{{c}}, ~18/11 = ~~854~~.~~633~~{{c}}, ~7/4 = 969.~~032~~{{c}}

* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}

* CWE: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~411~~{{c}}, ~18/11 = ~~854~~.~~634~~{{c}}, ~7/4 = 969.~~026~~{{c}}

: [[error map]]: {{val| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3927{{c}}, ~84/65 = 442.4436{{c}}, ~7/4 = 969.1894{{c}}

{{~~Optimal ET sequence~~|~~legend=~~0~~| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}~~

: error map: {{val| 0.000 +0.438 +0.966 +0.364 +1.453 +1.331 }}

~~Badness (Smith): 34~~.0 ~~× 10-6~~

~~==== 19-limit ====~~

~~Subgroup: 2~~.3.~~5.7~~.11.13.~~17.19~~

~~Comma list: 715/714~~, ~~1089/1088~~, ~~1216/1215~~, ~~1445/1444~~

{{Optimal ET sequence|legend=1| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}

~~Mapping: {{mapping| 1 0 2 0 1 -6 -4 -4 | 0 1 2 0 2 3 6 7 | 0 0 -4 0 -1 3 -2 -4 | 0 0 0 1 0 1 0 0 }}~~

[[Badness]] (Sintel): 1.13

~~Optimal tunings:~~

* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.368{{c}}, ~18/11 = 854.639{{c}}, ~7/4 = 969.075{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.351{{c}}, ~18/11 = 854.644{{c}}, ~7/4 = 969.010{{c}}

~~{{Optimal ET sequence|legend=0| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}~~

~~Badness (Smith): 41.9 × 10-6~~

~~== Semiporwellismic (16384/16335) ==~~

[[~~Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 16384/16335~~

~~{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 0 -2 | 0 0 2 0 -1 | 0 0 0 1 0 }}~~

~~: Mapping generators: ~2, ~3, ~128/99, ~7~~

~~[[Optimal tuning~~]] (~~[[CTE]]~~): ~~~2 = 1200.000{{c}}, ~3/2 = 702.470{{c}}, ~128/99 = 442.106{{c}}, ~7/4 = 968.826{{c}}~~

~~{{Optimal ET sequence|legend=~~1~~| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}~~

~~[[Badness]] (Smith): 0~~.~~219 × 10-6~~

== Symbiotic (19712/19683) ==

Line 816:

Line 880:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~{{c}}, ~3/2 = 702.~~213~~{{c}}, ~5/4 = 386.~~314~~{{c}}, ~7/4 = 968.~~736~~{{c}}

* [[WE]]: ~2 = 1199.9290{{c}}, ~3/2 = 702.2265{{c}}, ~5/4 = 386.4556{{c}}, ~7/4 = 968.8979{{c}}

* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~244~~{{c}}, ~5/4 = 386.~~406~~{{c}}, ~7/4 = 968.~~859~~{{c}}

: [[error map]]: {{val| -0.071 +0.201 -0.000 -0.070 -0.106 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2440{{c}}, ~5/4 = 386.4064{{c}}, ~7/4 = 968.8593{{c}}

: error map: {{val| 0.000 +0.289 +0.093 +0.033 +0.019 }}

{{Optimal ET sequence|legend=1| 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b }}

[[Badness]] (~~Smith~~): 0.~~120 × 10-6~~

[[Badness]] (Sintel): 2.19

=== Tridecimal symbiotic ===

Line 831:

Line 897:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~{{c}}, ~3/2 = 702.~~228~~{{c}}, ~5/4 = 386.~~284~~{{c}}, ~7/4 = 968.~~789~~{{c}}

* [[WE]]: ~2 = 1199.9268{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.4554{{c}}, ~7/4 = 968.9114{{c}}

* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~254~~{{c}}, ~5/4 = 386.~~401~~{{c}}, ~7/4 = 968.~~898~~{{c}}

: [[error map]]: {{val| -0.073 +0.201 -0.005 -0.061 -0.093 -0.012 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2544{{c}}, ~5/4 = 386.4007{{c}}, ~7/4 = 968.8979{{c}}

: error map: {{val| 0.000 +0.299 +0.087 +0.072 +0.074 +0.125 }}

{{Optimal ET sequence|legend=1| 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf }}

[[Badness]] (~~Smith~~): 3.~~31 × 10-6~~

[[Badness]] (Sintel): 0.868

=== 2.3.5.7.11.13.19 subgroup ===

Line 846:

Line 914:

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~{{c}}, ~3/2 = 702.~~230~~{{c}}, ~5/4 = 386.~~304~~{{c}}, ~7/4 = 968.~~797~~{{c}}

* WE: ~2 = 1199.9273{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.4508{{c}}, ~7/4 = 968.9092{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.~~255~~{{c}}, ~5/4 = 386.~~408~~{{c}}, ~7/4 = 968.~~902~~{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.2554{{c}}, ~5/4 = 386.4080{{c}}, ~7/4 = 968.9015{{c}}

{{Optimal ET sequence|legend=0| 17ch, 24f, 34dffh, 36ceh, 41, 53, 58h, 94, 111, 152f, 212h, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bfh }}

Badness (~~Smith~~): 4.~~37 × 10~~-6</~~sup>~~

Badness (Sintel): 0.375

== Argyric (41503/41472) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: [[41503/41472]]

{{Mapping|legend=1| 1 0 0 1 3 | 0 1 0 0 2 | 0 0 1 0 0 | 0 0 0 2 -3 }}

: Mapping generators: ~2, ~3, ~5, ~144/77

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0485{{c}}, ~3/2 = 701.9606{{c}}, ~5/4 = 386.2167{{c}}, ~144/77 = 1084.3250{{c}}

: [[error map]]: {{val| +0.049 +0.054 -0.000 -0.127 -0.129 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9739{{c}}, ~5/4 = 386.2460{{c}}, ~144/77 = 1084.2953{{c}}

: error map: {{val| 0.000 +0.019 -0.068 -0.235 -0.256 }}

{{Optimal ET sequence|legend=1| 19, 22, 31, 41, 53, 72, 166, 167, 176, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}

[[Badness]] (Sintel): 0.777

== Chrysic (43923/43904) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: [[43923/43904]]

{{Mapping|legend=1| 1 0 0 3 4 | 0 1 0 3 2 | 0 0 1 0 0 | 0 0 0 -4 -3 }}

: Mapping generators: ~2, ~3, ~5, ~33/14

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0167{{c}}, ~3/2 = 701.9323{{c}}, ~5/4 = 386.2803{{c}}, ~33/28 = 284.2370{{c}}

: [[error map]]: {{val| +0.017 -0.006 -0.000 +0.056 -0.114 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9341{{c}}, ~5/4 = 386.2927{{c}}, ~33/28 = 284.2352{{c}}

: error map: {{val| 0.000 -0.021 -0.021 +0.036 -0.155 }}

{{Optimal ET sequence|legend=1| 17c, 26, 29, 34d, 43, 46, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 2715, 2985, 3327e, 3479, 3821, 4091e, 4433e, 5539e }}

[[Badness]] (Sintel): 0.770

== Elysic (117649/117612) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: [[117649/117612]]

{{Mapping|legend=1| 1 0 0 0 -1 | 0 2 0 0 -5 | 0 0 1 0 0 | 0 0 0 1 3 }}

: Mapping generators: ~2, ~343/198, ~5, ~7

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0027{{c}}, ~343/198 = 950.9861{{c}}, ~5/4 = 386.3082{{c}}, ~7/4 = 968.7558{{c}}

: [[error map]]: {{val| +0.003 +0.017 -0.000 -0.065 +0.033 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/198 = 950.9850{{c}}, ~5/4 = 386.3105{{c}}, ~7/4 = 968.7571{{c}}

: error map: {{val| 0.000 +0.015 -0.003 -0.069 +0.028 }}

{{Optimal ET sequence|legend=1| 49, 58, 72, 130, 193, 198, 212, 270, 342, 612, 954, 1084, 1354, 1547, 1817, 2159, 3243, 3513, 5060, 5402, 5672, 8915d, 11074d }}

[[Badness]] (Sintel): 0.901

== Olympic (131072/130977) ==

Olympic is highly accurate yet easy to conceptualize and notate on the staff. It equates [[33/32]] with ([[64/63]])2, and [[36/35]] with [[1053/1024]], reducing the four [[formal comma]]s required for the [[13-limit]] to two. Therefore it can be notated in [[chain-of-fifths notation]] with two additional set of accidentals, one for the septimal comma step, and the other for the [[aberschisma]] step. It can be viewed as the [[detemperament]] of [[cassaschismic]] that decouples the Pythagorean and septimal commas.

It also has a [[strong extension]] to include [[29/1|prime 29]] though with higher complexity.

[[Subgroup]]: 2.3.5.7.11

Line 861:

Line 987:

: Mapping generators: ~2, ~3, ~5, ~7

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~103~~{{c}}, ~5/4 = 386.~~314~~{{c}}, ~7/4 = 969.~~012~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9460{{c}}, ~3/2 = 702.0489{{c}}, ~5/4 = 386.4217{{c}}, ~7/4 = 968.9839{{c}}

: [[error map]]: {{val| -0.054 +0.040 -0.000 +0.050 +0.038 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0853{{c}}, ~5/4 = 386.4125{{c}}, ~7/4 = 969.0242{{c}}

: error map: {{val| 0.000 +0.130 +0.099 +0.198 +0.207 }}

{{Optimal ET sequence|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}

[[Badness]] (Sintel): 2.05

=== Tridecimal olympic ===

==== 13-limit ====

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 2080/2079, 4096/4095

[[Comma list]]: [[2080/2079]], [[4096/4095]]

{{Mapping|legend=1| 1 0 0 0 17 12 | 0 1 0 0 -5 -2 | 0 0 1 0 0 -1 | 0 0 0 1 -2 -1 }}

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~103~~{{c}}, ~5/4 = 386.~~298~~{{c}}, ~7/4 = 969.~~007~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9638{{c}}, ~3/2 = 702.0645{{c}}, ~5/4 = 386.3208{{c}}, ~7/4 = 968.9712{{c}}

: [[error map]]: {{val| -0.036 +0.073 -0.065 +0.073 +0.128 -0.166 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0884{{c}}, ~5/4 = 386.3269{{c}}, ~7/4 = 969.0015{{c}}

: error map: {{val| 0.000 +0.133 +0.013 +0.176 +0.237 -0.033 }}

{{Optimal ET sequence|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }} *

<nowiki>*</nowiki> [[optimal patent val]]: [[3044edo|3044]]

[[Badness]] (Sintel): 0.292

===== 2.3.5.7.11.13.29 subgroup =====

Subgroup: 2.3.5.7.11.13.29

Comma list: 2080/2079, 3510/3509, 4096/4095

Mapping: {{mapping| 1 0 0 0 17 12 -21 | 0 1 0 0 -5 -2 11 | 0 0 1 0 0 -1 0 | 0 0 0 1 -2 -1 3 }}

Optimal tunings:

* WE: ~2 = 1199.960{{c}}, ~3/2 = 702.008{{c}}, ~5/4 = 386.247{{c}}, ~7/4 = 968.895{{c}}

: error map: {{val| -0.040 +0.053 -0.067 +0.069 +0.164 -0.171 +0.048 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.070{{c}}, ~5/4 = 386.337{{c}}, ~7/4 = 969.013{{c}}

: error map: {{val| 0.000 +0.115 +0.023 +0.187 +0.304 -0.018 +0.237 }}

{{Optimal ET sequence|legend=0| 41, 53j, 87, 130j, 176, 183, 217, 224, 270, 311, 487, 494, 1205, 1699 }}

Badness (Sintel): 0.505

==== Commas 2080/2079, 3250/3249, 4096/4095 ====

This is a [[weak extension]] of olympic, requiring the split of the prime 7 in half; or alternatively, the septimal comma or the garischisma.

Subgroup: 2.3.5.7.11.13.19

Comma list: 2080/2079, 3250/3249, 4096/4095

Mapping: {{mapping| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}

Optimal tunings:

* WE: ~2 = 1199.9639{{c}}, ~3/2 = 702.0643{{c}}, ~5/4 = 386.3218{{c}}, ~247/132 = 1084.4676{{c}}

: error map: {{val| -0.036 +0.073 -0.064 +0.073 +0.129 -0.166 -0.004 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0887{{c}}, ~5/4 = 386.3257{{c}}, ~247/132 = 1084.5006{{c}}

: error map: {{val| 0.000 +0.134 +0.012 +0.175 +0.236 -0.032 +0.135 }}

{{Optimal ET sequence|legend=0| 41, 53, 84, 94, 137, 176, 217, 270, 581, 851, 935, 1205, 2193, 2463, 2774e }}

Badness (Sintel): 0.618

===== 2.3.5.7.11.13.19.29 subgroup =====

Subgroup: 2.3.5.7.11.13.19.29

Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095

Mapping: {{mapping| 1 0 0 1 15 11 -13 6 -18 | 0 1 0 0 -5 -2 5 -2 -11 | 0 0 1 0 0 -1 2 1 0 | 0 0 0 2 -4 -2 5 -1 6 }}

Optimal tunings:

* WE: ~2 = 1199.9594{{c}}, ~3/2 = 702.0487{{c}}, ~5/4 = 386.3262{{c}}, ~247/132 = 1084.4677{{c}}

: error map: {{val| -0.041 +0.053 -0.069 +0.069 +0.163 -0.170 +0.005 +0.048 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0709{{c}}, ~5/4 = 386.3327{{c}}, ~247/132 = 1084.5064{{c}}

: error map: {{val| -0.000 +0.116 +0.019 +0.187 +0.302 -0.015 +0.171 +0.241 }}

{{Optimal ET sequence|legend=0| 41, 53j, 94, 137j, 176, 217, 270, 487, 581j, 718, 894, 935, 988, 1205, 2193, 3398dehj}}

Badness (Sintel): 1.10

==== Metaolympic ====

Metaolympic is the above extension with prime 17 added. It is better without prime 17, but prime 17 can be added easily into the temperament by also tempering out 4200/4199, which also takes advantage of the split prime 7 at the cost of higher complexity.

Subgroup: 2.3.5.7.11.13.17

Comma list: 2080/2079, 4096/4095, 35000/34969

Mapping: {{mapping| 1 0 0 1 15 11 -13 | 0 1 0 0 -5 -2 5 | 0 0 1 0 0 -1 2 | 0 0 0 2 -4 -2 5 }}

: mapping generators: ~2, ~3, ~5, ~187/100

Optimal tunings:

* WE: ~2 = 1199.9530{{c}}, ~3/2 = 702.0493{{c}}, ~5/4 = 386.2541{{c}}, ~187/100 = 1084.4393{{c}}

: error map: {{val| -0.047 +0.047 -0.154 +0.006 +0.208 -0.088 +0.184 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0798{{c}}, ~5/4 = 386.2658{{c}}, ~187/100 = 1084.4811{{c}}

: error map: {{val| 0.000 +0.125 -0.057 +0.136 +0.359 +0.094 +0.363 }}

{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2815eg }}

Badness (Sintel): 1.59

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199

Mapping: {{mapping| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}

Optimal tunings:

* WE: ~2 = 1199.9539{{c}}, ~3/2 = 702.0464{{c}}, ~5/4 = 386.2644{{c}}, ~187/100 = 1084.4405{{c}}

: error map: {{val| -0.046 +0.045 -0.141 +0.009 +0.227 -0.088 0.192 -0.058 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0768{{c}}, ~5/4 = 386.2655{{c}}, ~187/100 = 1084.4813{{c}}

: error map: {{val| 0.000 +0.122 -0.048 +0.137 +0.373 +0.090 +0.367 +0.118 }}

{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2451eg }}

Badness (Sintel): 0.764

===== 2.3.5.7.11.13.17.19.29 subgroup =====

Subgroup: 2.3.5.7.11.13.17.19.29

Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095, 4200/4199

Mapping: {{mapping| 1 0 0 1 15 11 -13 6 -18 | 0 1 0 0 -5 -2 5 -2 -11 | 0 0 1 0 0 -1 2 1 0 | 0 0 0 2 -4 -2 5 -1 6 }}

Optimal tunings:

* WE: ~2 = 1199.9561{{c}}, ~3/2 = 702.0552{{c}}, ~5/4 = 386.2577{{c}}, ~187/100 = 1084.4387{{c}}

: error map: {{val| -0.044 +0.056 -0.144 +0.008 +0.212 -0.080 +0.205 -0.068 -0.031 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0795{{c}}, ~5/4 = 386.2631{{c}}, ~187/100 = 1084.4800{{c}}

: error map: {{val| 0.000 +0.124 -0.051 +0.134 +0.365 +0.090 +0.368 +0.111 +0.177 }}

{{Optimal ET sequence|legend=0| 53j, 94, 137j, 176g, 217, 270, 311, 487, 581j, 718, 935, 1205g, 1516e, 2234eg, 2721egg }}

Badness (Sintel): 1.27

== Alphaxenic (131769/131072) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 131769/131072

{{Mapping|legend=1| 2 1 0 0 8 | 0 2 0 0 -1 | 0 0 1 0 0 | 0 0 0 1 0 }}

: mapping generators: ~363/256, ~16/11, ~5, ~7

[[Optimal tuning]]s:

* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}, ~5/4 = 385.6773{{c}}, ~7/4 = 968.1894{{c}}

: [[error map]]: {{val| +0.318 -0.094 -0.000 -0.000 -0.897 }}

* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}, ~5/4 = 385.7900{{c}}, ~7/4 = 968.1927{{c}}

: error map: {{val| 0.000 -0.491 -0.524 -0.633 -2.050 }}

{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}

[[Badness]] (Sintel): 23.0

== Seascape (160083/160000) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: [[~~Seascape comma|~~160083/160000]]

[[Comma list]]: [[160083/160000]]

{{Mapping|legend=1| 1 0 0 0 4 | 0 2 0 0 -3 | 0 0 1 0 2 | 0 0 0 1 -1 }}

: Mapping generators: ~2, ~400/231, ~5, ~7

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.~~000~~{{c}}, ~400/231 = 950.~~960~~{{c}}, ~5/4 = 386.~~417~~{{c}}, ~7/4 = 968.~~751~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0285{{c}}, ~400/231 = 950.9641{{c}}, ~5/4 = 386.3335{{c}}, ~7/4 = 968.7129{{c}}

: [[error map]]: {{val| +0.028 -0.027 +0.077 -0.056 -0.085 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 950.9492{{c}}, ~5/4 = 386.3638{{c}}, ~7/4 = 969.7159{{c}}

: error map: {{val| 0.000 -0.057 +0.050 -0.110 -0.154 }}

{{Optimal ET sequence|legend=1| 14c, 15, 19, 29, 39d, 43, 53, 58, 72, 130, 183, 198, 212, 255, 270, 342, 612, 954, 1084, 1354, 1696, 2833e, 3575, 3917e, 4187e, 4529de, 5883de }}

~~{{Optimal ET sequence|legend=1| 15, 19, 29, 43, 53, 58, 68, 72 }}~~

[[Badness]] (Sintel): 0.596

== Fifthchromismic (1610510/1594323) ==

Line 895:

Line 1,165:

[[Comma list]]: [[1610510/1594323]]

{{Mapping|legend=1| 1 0 4 0 -1 | 0 1 3 0 2 | 0 0 -5 0 1 | 0 0 0 1 0 }}

{{Mapping|legend=1| 1 0 -6 0 1 | 0 1 3 0 2 | 0 0 5 0 -1 | 0 0 0 1 0 }}

: Mapping generators: ~2, ~3, ~22/9, ~7

: Mapping generators: ~2, ~3, ~18/11, ~7

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~738~~{{c}}, ~11/9 = ~~344~~.~~406~~{{c}}, ~7/4 = 968.~~826~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9757{{c}}, ~3/2 = 702.7610{{c}}, ~18/11 = 855.5656{{c}}, ~7/4 = 968.8735{{c}}

: [[error map]]: {{val| -0.024 +0.782 -0.130 -0.001 -1.434 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.7565{{c}}, ~18/11 = 855.5885{{c}}, ~7/4 = 968.8591{{c}}

: error map: {{val| 0.000 +0.801 -0.102 +0.033 -1.393 }}

{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 251, 258e, 338, 432, 519 }}

[[Badness]] (Sintel): 97.2

=== Fifthchroma ===

[[Subgroup]]: 2.3.5.7.11.13

Line 909:

Line 1,185:

[[Comma list]]: 352/351, 951665/944784

{{Mapping|legend=1| 1 0 4 0 -1 4 | 0 1 3 0 2 -1 | 0 0 -5 0 1 1 | 0 0 0 1 0 0 }}

{{Mapping|legend=1| 1 0 -6 0 1 6 | 0 1 3 0 2 -1 | 0 0 5 0 -1 -1 | 0 0 0 1 0 0 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8431{{c}}, ~3/2 = 702.9005{{c}}, ~18/11 = 855.4090{{c}}, ~7/4 = 969.1386{{c}}

: [[error map]]: {{val| -0.157 +0.789 -0.097 -0.001 -1.396 +0.379 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.9326{{c}}, ~18/11 = 855.5400{{c}}, ~7/4 = 969.1437{{c}}

: error map: {{val| 0.000 +0.978 +0.184 +0.318 -0.993 +1.000 }}

[[Optimal ~~tuning]] ([[CTE]]): ~2~~ = ~~1200.000{{c}}~~, ~~~3/2 = 702.776{{c}}~~, ~~~11/9 = 344.410{{c}}~~, ~~~7/4 = 968.826{{c~~}}

{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 181, 244, 251, 261, 338, 599b }}

~~{{Optimal ET sequence|legend=1| 77, 80, 87, 94, 167 }}~~

[[Badness]] (Sintel): 7.63

==== 17-limit ====

Line 920:

Line 1,202:

Comma list: 352/351, 715/714, 155771/155520

Mapping: {{mapping| 1 0 4 0 -1 4 6 | 0 1 3 0 2 -1 3 | 0 0 -5 0 1 1 -3 | 0 0 0 1 0 0 -1 }}

Mapping: {{mapping| 1 0 -6 0 1 6 0 | 0 1 3 0 2 -1 3 | 0 0 5 0 -1 -1 3 | 0 0 0 1 0 0 -1 }}

Optimal tunings:

* WE: ~2 = 1199.8126{{c}}, ~3/2 = 702.8981{{c}}, ~11/9 = 855.3764{{c}}, ~7/4 = 969.3543{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9376{{c}}, ~11/9 = 855.5306{{c}}, ~7/4 = 969.4313{{c}}

Optimal ~~tuning (CTE): ~2~~ = ~~1200.000{{c}}~~, ~~~3/2 = 702.713{{c}}~~, ~~~11/9 = 344.403{{c}}~~, ~~~7/4 = 969.193{{c~~}}

{{Optimal ET sequence|legend=0| 70, 73g, 77, 80, 87, 94, 157, 167, 181, 251, 261, 338, 432, 599bg }}

~~{{Optimal ET sequence|legend=0| 77, 80, 87, 94, 167 }}~~

Badness (Sintel): 3.65

==== 19-limit ====

Line 931:

Line 1,217:

Comma list: 352/351, 715/714, 1463/1458, 2057/2052

Mapping: {{mapping| 1 0 4 0 -1 4 6 2 | 0 1 3 0 2 -1 3 4 | 0 0 -5 0 1 1 -3 -1 | 0 0 0 1 0 0 -1 -1 }}

Mapping: {{mapping| 1 0 -6 0 1 6 0 0 | 0 1 3 0 2 -1 3 4 | 0 0 5 0 -1 -1 3 1 | 0 0 0 1 0 0 -1 -1 }}

Optimal tunings:

* WE: ~2 = 1199.8128{{c}}, ~3/2 = 702.9226{{c}}, ~11/9 = 855.3621{{c}}, ~7/4 = 969.3187{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9618{{c}}, ~11/9 = 855.5163{{c}}, ~7/4 = 969.3961{{c}}

{{Optimal ET sequence|legend=0| 70, 73gh, 77, 80, 87, 94, 157, 167h, 181, 251, 261, 338, 599bg }}

Badness (Sintel): 2.63

== Quartismic (117440512/117406179) ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 117440512/117406179

{{Mapping|legend=1| 1 0 0 1 5 | 0 1 0 1 -1 | 0 0 1 0 0 | 0 0 0 5 1 }}

: mapping generators: ~2, ~3, ~5, ~33/32

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9875{{c}}, ~3/2 = 701.9753{{c}}, ~5/4 = 386.3387{{c}}, ~33/32 = 53.3743{{c}}

: [[error map]]: {{val| -0.012 +0.008 -0.000 -0.004 +0.031 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9802{{c}}, ~5/4 = 386.3343{{c}}, ~33/32 = 53.3730{{c}}

: error map: {{val| 0.000 +0.025 +0.021 +0.019 +0.075 }}

{{Optimal ET sequence|legend=1| 21, 22, 43, 46, 65d, 68, 89, 111, 159, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2159, 2653, 3125, 3395, 7060, 7554, 10949e, 14614e, 15850ee, 22168bdee, 23404bcdee, 26799bcdeee, 34353bcdeeee }}

[[Badness]] (Sintel): 5.00

=== Tridecimal quartismic ===

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 6656/6655, 123201/123200

{{Mapping|legend=1| 1 0 0 1 5 6 | 0 1 0 1 -1 -3 | 0 0 1 0 0 1 | 0 0 0 5 1 3 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9884{{c}}, ~3/2 = 701.9674{{c}}, ~5/4 = 386.3432{{c}}, ~33/32 = 53.3763{{c}}

: [[error map]]: {{val| -0.012 +0.001 +0.006 -0.000 +0.045 -0.016 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9728{{c}}, ~5/4 = 386.3385{{c}}, ~33/32 = 53.3748{{c}}

: error map: {{val| 0.000 +0.018 +0.025 +0.021 +0.084 +0.017 }}

Optimal ~~tuning (CTE): ~2~~ = ~~1200.000{{c}}~~, ~~~3/2 = 702.736{{c}}~~, ~~~11/9 = 344.417{{c}}~~, ~~~7/4 = 969.~~159~~{{c~~}}

{{Optimal ET sequence|legend=1| 22, 43f, 46, 65d, 89f, 111, 159, 224, 270, 494, 764, 1012, 1236, 1506, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816dee }}

~~{{Optimal ET sequence|legend=~~0~~| 77, 80, 87, 94, 167h }}~~

[[Badness]] (Sintel): 0.456

~~[[Category:Temperaments]]~~

~~[[Category:Lists of temperaments]]~~

[[Category:Temperament collections]]

[[Category:~~Regular temperament theory]]~~

[[Category:Catalogs of rank-4 temperaments]]

~~[[Category:Rank~~ 4| ]] ~~~~

@@ Line 470: / Line 470: @@
 [[Badness]] (Sintel): 0.454
-=== Commas 540/539, 847/845 ===
-[[Subgroup]]: 2.3.5.7.11.13
-[[Comma list]]: 540/539, 847/845
-{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 0 0 3 3 | 0 0 1 1 -1 -1 | 0 0 0 2 -4 -3 }}
-: Mapping generators: ~2, ~3, ~5, ~13/11
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.8339{{c}}, ~3/2 = 701.4961{{c}}, ~5/4 = 386.5624{{c}}, ~13/11 = 291.6840{{c}}
-: [[error map]]: {{val| -0.166 -0.625 -0.084 +0.772 -0.627 +1.848 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4439{{c}}, ~5/4 = 386.4201{{c}}, ~13/11 = 291.7287{{c}}
-: error map: {{val| 0.000 -0.511 +0.106 +1.052 -0.321 +2.198 }}
-{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
-[[Badness]] (Sintel): 1.04
 === Commas 540/539, 625/624 ===
@@ Line 523: / Line 505: @@
 [[Badness]] (Sintel): 0.802
+=== Commas 540/539, 847/845 ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 540/539, 847/845
+{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 0 0 3 3 | 0 0 1 1 -1 -1 | 0 0 0 2 -4 -3 }}
+: Mapping generators: ~2, ~3, ~5, ~13/11
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8339{{c}}, ~3/2 = 701.4961{{c}}, ~5/4 = 386.5624{{c}}, ~13/11 = 291.6840{{c}}
+: [[error map]]: {{val| -0.166 -0.625 -0.084 +0.772 -0.627 +1.848 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4439{{c}}, ~5/4 = 386.4201{{c}}, ~13/11 = 291.7287{{c}}
+: error map: {{val| 0.000 -0.511 +0.106 +1.052 -0.321 +2.198 }}
+{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
+[[Badness]] (Sintel): 1.04
 == Pentacircle (896/891) ==
@@ Line 578: / Line 578: @@
 [[Badness]] (Sintel): 1.04
+== Moctdelismic (1375/1372) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 1375/1372
+{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 0 | 0 0 1 0 -3 | 0 0 0 1 3 }}
+: mapping generators: ~2, ~3, ~5, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0558{{c}}, ~3/2 = 701.8991{{c}}, ~5/4 = 385.7504{{c}}, ~7/4 = 969.3743{{c}}
+: [[error map]]: {{val| +0.056 -0.000 -0.452 +0.660 -0.334 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9100{{c}}, ~5/4 = 385.7894{{c}}, ~7/4 = 969.4163{{c}}
+: error map: {{val| 0.000 -0.045 -0.524 +0.590 -0.437 }}
+{{Optimal ET sequence|legend=1| 12, 19e, 27, 29, 31, 41, 60e, 68, 72, 140, 152, 224, 364, 436, 547, 588, 619, 843, 1055ce }}
+[[Badness]] (Sintel): 0.723
 == Lehmerismic (3025/3024) ==
@@ Line 613: / Line 631: @@
 [[Badness]] (Sintel): 0.272
+== Vishdelismic (5632/5625) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 5632/5625
+{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 0 | 0 0 1 0 -3 | 0 0 0 1 3 }}
+: mapping generators: ~2, ~3, ~5, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8976{{c}}, ~3/2 = 702.1145{{c}}, ~5/4 = 386.7638{{c}}, ~7/4 = 969.0306{{c}}
+: [[error map]]: {{val| -0.102 +0.057 +0.245 -0.000 -0.136 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1309{{c}}, ~5/4 = 386.7770{{c}}, ~7/4 = 969.0087{{c}}
+: error map: {{val| 0.000 +0.176 +0.463 +0.183 +0.052 }}
+{{Optimal ET sequence|legend=1| 12, 19e, 22, 31, 53, 84e, 87, 99e, 118, 130, 152, 239, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}
+[[Badness]] (Sintel): 0.804
 == Trimitone (8019/8000) ==
@@ Line 638: / Line 674: @@
 {{Mapping|legend=1| 1 0 0 0 6 -3 | 0 1 0 0 -6 6 | 0 0 1 0 3 0 | 0 0 0 1 0 -1 }}
-: Mapping generators: ~2, ~3, ~5, ~7
 [[Optimal tuning]]s:
@@ Line 649: / Line 684: @@
 [[Badness]] (Sintel): 0.875
+=== Commas 351/350, 2080/2079 ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 351/350, 2080/2079
+{{Mapping|legend=1| 1 0 0 0 6 1 | 0 1 0 0 -6 -3 | 0 0 1 0 3 2 | 0 0 0 1 0 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9538{{c}}, ~3/2 = 701.5852{{c}}, ~5/4 = 387.1356{{c}}, ~7/4 = 969.7495{{c}}
+: [[error map]]: {{val| -0.046 -0.416 +0.730 +0.831 +0.301 -1.447 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.5936{{c}}, ~5/4 = 387.0975{{c}}, ~7/4 = 969.7451{{c}}
+: error map: {{val| 0.000 -0.361 +0.784 +0.919 +0.413 -1.369 }}
+{{Optimal ET sequence|legend=1| 19, 26, 39df, 46, 53, 58, 72, 111, 130, 183, 313 }}
+[[Badness]] (Sintel): 0.677
 == Kalismic (9801/9800) ==
@@ Line 668: / Line 720: @@
 [[Badness]] (Sintel): 0.0415
-=== Commas 1716/1715, 2080/2079 ===
+=== Briggs ===
+Briggs tempers out [[1716/1715]] and [[2080/2079]], and is the join of [[pessoal]] and [[greenland]], among many others.
 [[Subgroup]]: 2.3.5.7.11.13
@@ Line 691: / Line 745: @@
 [[Badness]] (Sintel): 0.256
+=== Kalimera ===
+Kalimera is a [[weak extension]] of kalismic by adding the [[harmonisma]], and also an extremely high accuracy detemperament of [[donar]] achieved by observing the [[ragisma]].
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 9801/9800, 10648/10647
+{{Mapping|legend=1| 2 0 1 0 4 9 | 0 1 0 0 -2 -4 | 0 0 2 0 2 3 | 0 0 0 1 1 1 }}
+: mapping generators: ~99/70, ~3, ~1053/560, ~7
+[[Optimal tuning]]:
+* [[WE]]: ~99/70 = 600.0006{{c}}, ~3/2 = 701.9470{{c}}, ~1053/560 = 1093.1617{{c}}, ~7/4 = 968.8431{{c}}
+: [[error map]]: {{val| +0.001 -0.007 +0.010 +0.020 -0.043 +0.015 }}
+* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9469{{c}}, ~1053/560 = 1093.1614{{c}}, ~7/4 = 968.8438{{c}}
+: error map: {{val| 0.000 -0.008 +0.009 +0.018 -0.045 +0.013 }}
+{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 104c, 132de, 144, 166, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 5144, 6380, 7828, 9334, 12018 }}
+[[Badness]] (Sintel): 0.164
 == Unisquary (12005/11979) ==
@@ Line 697: / Line 771: @@
 [[Comma list]]: [[12005/11979]]
-{{Mapping|legend=1| 1 0 0 0 0 | 0 1 0 2 2 | 0 0 1 -1 -1 | 0 0 0 3 4 }}
+{{Mapping|legend=1| 1 0 0 -6 -8 | 0 1 0 2 2 | 0 0 1 2 3 | 0 0 0 3 4 }}
-: Mapping generators: ~2, ~3, ~5, ~11/7
+: Mapping generators: ~2, ~3, ~5, ~44/35
 [[Optimal tuning]]s:
-* [[CTE]] and [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.031{{c}}, ~5/4 = 386.233{{c}}, ~11/7 = 783.508{{c}}
+* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0306{{c}}, ~5/4 = 386.2325{{c}}, ~44/35 = 397.2749{{c}}
+: [[error map]]: {{val| -0.000 +0.076 -0.081 +0.243 -0.475 +0.541 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0306{{c}}, ~5/4 = 386.2325{{c}}, ~44/35 = 397.2750{{c}}
+: error map: {{val| 0.000 +0.076 -0.081 -0.475 +0.541 }}
 {{Optimal ET sequence|legend=1| 12, 21, 23de, 26, 37, 46, 58, 72, 118, 130, 190, 239, 311, 441, 559, 752e, 870 }}
+[[Badness]] (Sintel): 2.43
 === Hensquary ===
@@ Line 710: / Line 789: @@
 [[Comma list]]: 364/363, 1716/1715
-{{Mapping|legend=1| 1 0 0 0 0 -2 | 0 1 0 2 2 3 | 0 0 1 -1 -1 -1 | 0 0 0 3 4 5 }}
+{{Mapping|legend=1| 1 0 0 -6 -8 -12 | 0 1 0 2 2 3 | 0 0 1 2 3 4 | 0 0 0 3 4 5 }}
-: Mapping generators: ~2, ~3, ~5, ~11/7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.072{{c}}, ~5/4 = 386.528{{c}}, ~11/7 = 783.823{{c}}
+* [[WE]]: ~2 = 1199.7826{{c}}, ~3/2 = 702.2831{{c}}, ~5/4 = 386.9208{{c}}, ~44/35 = 396.8581{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.253{{c}}, ~5/4 = 386.776{{c}}, ~11/7 = 783.875{{c}}
+: [[error map]]: {{val| -0.217 +0.111 +0.172 +0.156 +1.443 -1.487 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2535{{c}}, ~5/4 = 386.7762{{c}}, ~44/35 = 397.0993{{c}}
+: error map: {{val| 0.000 +0.298 +0.463 +0.531 +1.915 -1.166 }}
 {{Optimal ET sequence|legend=1| 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369 }}
+[[Badness]] (Sintel): 0.988
 === Ekasquary ===
@@ Line 724: / Line 806: @@
 [[Comma list]]: 1575/1573, 4459/4455
-{{Mapping|legend=1| 1 0 0 0 0 0 | 0 1 0 2 2 0 | 0 0 1 -1 -1 3 | 0 0 0 3 4 -5 }}
+{{Mapping|legend=1| 1 0 0 -6 -8 10 | 0 1 0 2 2 0 | 0 0 1 2 3 -2 | 0 0 0 3 4 -5 }}
-: Mapping generators: ~2, ~3, ~5, ~11/7
 [[Optimal tuning]]s:
-* [[CTE]] and [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.952{{c}}, ~5/4 = 386.139{{c}}, ~11/7 = 783.548{{c}}
+* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.9524{{c}}, ~5/4 = 386.1388{{c}}, ~44/35 = 397.4088{{c}}
+: [[error map]]: {{val| -0.000 -0.003 -0.175 -0.417 +0.639 +0.150 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9524{{c}}, ~5/4 = 386.1388{{c}}, ~44/35 = 397.4089{{c}}
+: error map: {{val| 0.000 -0.003 -0.175 -0.417 +0.639 +0.150 }}
 {{Optimal ET sequence|legend=1| 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072 }}
-== Semicanousmic (14641/14580) ==
+[[Badness]] (Sintel): 1.12
+== Semicathartismic (14641/14580) ==
 [[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[14641/14580]]
-{{Mapping|legend=1| 1 0 2 0 1 | 0 1 2 0 2 | 0 0 -4 0 -1 | 0 0 0 1 0 }}
+{{Mapping|legend=1| 1 0 -2 0 0 | 0 1 2 0 2 | 0 0 4 0 1 | 0 0 0 1 0 }}
-: Mapping generators: ~2, ~3, ~18/11, ~7
+: Mapping generators: ~2, ~3, ~11/9, ~7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.334{{c}}, ~18/11 = 854.555{{c}}, ~7/4 = 968.826{{c}}
+* [[WE]]: ~2 = 1200.0495{{c}}, ~3/2 = 702.2793{{c}}, ~11/9 = 345.4721{{c}}, ~7/4 = 968.7265{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.292{{c}}, ~18/11 = 854.548{{c}}, ~7/4 = 968.756{{c}}
+: [[error map]]: {{val| +0.049 +0.374 +0.133 -0.000 -1.188 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2918{{c}}, ~11/9 = 345.4516{{c}}, ~7/4 = 968.7556{{c}}
+: error map: {{val| 0.000 +0.337 +0.077 -0.070 -1.283 }}
 {{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
-[[Badness]] (Smith): 0.351 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 6.40
-=== Tridecimal semicanousmic ===
+== Semiporwellismic (16384/16335) ==
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11
-Comma list: 2080/2079, 14641/14580
+[[Comma list]]: 16384/16335
-Mapping: {{mapping| 1 0 2 0 1 -6 | 0 1 2 0 2 3 | 0 0 -4 0 -1 3 | 0 0 0 1 0 1 }}
+{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 0 -2 | 0 0 2 0 -1 | 0 0 0 1 0 }}
+: Mapping generators: ~2, ~3, ~128/99, ~7
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.469{{c}}, ~18/11 = 854.644{{c}}, ~7/4 = 968.959{{c}}
+* [[WE]]: ~2 = 1199.7329{{c}}, ~3/2 = 702.3657{{c}}, ~128/99 = 442.1589{{c}}, ~7/4 = 969.3599{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.483{{c}}, ~18/11 = 854.642{{c}}, ~7/4 = 968.989{{c}}
+: [[error map]]: {{val| -0.267 +0.144 +0.103 -0.000 +0.456 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5107{{c}}, ~128/99 = 442.2243{{c}}, ~7/4 = 969.4120{{c}}
+: error map: {{val| 0.000 +0.556 +0.646 +0.586 +1.436 }}
-{{Optimal ET sequence|legend=0| 87, 111, 181, 198, 323, 410 }}
+{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}
-Badness (Smith): 17.1 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 4.00
-==== 17-limit ====
+=== Tridecimal semiporwellismic ===
-Subgroup: 2.3.5.7.11.13.17
+[[Subgroup]]: 2.3.5.7.11.13
-Comma list: 715/714, 1089/1088, 14641/14580
+[[Comma list]]: 2080/2079, 3584/3575
-Mapping: {{mapping| 1 0 2 0 1 -6 -4 | 0 1 2 0 2 3 6 | 0 0 -4 0 -1 3 -2 | 0 0 0 1 0 1 0 }}
+{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 1 0 -2 0 | 0 0 2 0 -1 -3 | 0 0 0 1 0 1 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.413{{c}}, ~18/11 = 854.633{{c}}, ~7/4 = 969.032{{c}}
+* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.411{{c}}, ~18/11 = 854.634{{c}}, ~7/4 = 969.026{{c}}
+: [[error map]]: {{val| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3927{{c}}, ~84/65 = 442.4436{{c}}, ~7/4 = 969.1894{{c}}
-{{Optimal ET sequence|legend=0| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
+: error map: {{val| 0.000 +0.438 +0.966 +0.364 +1.453 +1.331 }}
-Badness (Smith): 34.0 × 10<sup>-6</sup>
-==== 19-limit ====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444
+{{Optimal ET sequence|legend=1| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}
-Mapping: {{mapping| 1 0 2 0 1 -6 -4 -4 | 0 1 2 0 2 3 6 7 | 0 0 -4 0 -1 3 -2 -4 | 0 0 0 1 0 1 0 0 }}
+[[Badness]] (Sintel): 1.13
-Optimal tunings:
-* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.368{{c}}, ~18/11 = 854.639{{c}}, ~7/4 = 969.075{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.351{{c}}, ~18/11 = 854.644{{c}}, ~7/4 = 969.010{{c}}
-{{Optimal ET sequence|legend=0| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
-Badness (Smith): 41.9 × 10<sup>-6</sup>
-== Semiporwellismic (16384/16335) ==
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 16384/16335
-{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 0 -2 | 0 0 2 0 -1 | 0 0 0 1 0 }}
-: Mapping generators: ~2, ~3, ~128/99, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.470{{c}}, ~128/99 = 442.106{{c}}, ~7/4 = 968.826{{c}}
-{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}
-[[Badness]] (Smith): 0.219 × 10<sup>-6</sup>
 == Symbiotic (19712/19683) ==
@@ Line 816: / Line 880: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 968.736{{c}}
+* [[WE]]: ~2 = 1199.9290{{c}}, ~3/2 = 702.2265{{c}}, ~5/4 = 386.4556{{c}}, ~7/4 = 968.8979{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.244{{c}}, ~5/4 = 386.406{{c}}, ~7/4 = 968.859{{c}}
+: [[error map]]: {{val| -0.071 +0.201 -0.000 -0.070 -0.106 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2440{{c}}, ~5/4 = 386.4064{{c}}, ~7/4 = 968.8593{{c}}
+: error map: {{val| 0.000 +0.289 +0.093 +0.033 +0.019 }}
 {{Optimal ET sequence|legend=1| 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b }}
-[[Badness]] (Smith): 0.120 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 2.19
 === Tridecimal symbiotic ===
@@ Line 831: / Line 897: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.228{{c}}, ~5/4 = 386.284{{c}}, ~7/4 = 968.789{{c}}
+* [[WE]]: ~2 = 1199.9268{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.4554{{c}}, ~7/4 = 968.9114{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.254{{c}}, ~5/4 = 386.401{{c}}, ~7/4 = 968.898{{c}}
+: [[error map]]: {{val| -0.073 +0.201 -0.005 -0.061 -0.093 -0.012 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2544{{c}}, ~5/4 = 386.4007{{c}}, ~7/4 = 968.8979{{c}}
+: error map: {{val| 0.000 +0.299 +0.087 +0.072 +0.074 +0.125 }}
 {{Optimal ET sequence|legend=1| 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf }}
-[[Badness]] (Smith): 3.31 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.868
 === 2.3.5.7.11.13.19 subgroup ===
@@ Line 846: / Line 914: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.230{{c}}, ~5/4 = 386.304{{c}}, ~7/4 = 968.797{{c}}
+* WE: ~2 = 1199.9273{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.4508{{c}}, ~7/4 = 968.9092{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.255{{c}}, ~5/4 = 386.408{{c}}, ~7/4 = 968.902{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.2554{{c}}, ~5/4 = 386.4080{{c}}, ~7/4 = 968.9015{{c}}
 {{Optimal ET sequence|legend=0| 17ch, 24f, 34dffh, 36ceh, 41, 53, 58h, 94, 111, 152f, 212h, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bfh }}
-Badness (Smith): 4.37 × 10<sup>-6</sup>
+Badness (Sintel): 0.375
+== Argyric (41503/41472) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: [[41503/41472]]
+{{Mapping|legend=1| 1 0 0 1 3 | 0 1 0 0 2 | 0 0 1 0 0 | 0 0 0 2 -3 }}
+: Mapping generators: ~2, ~3, ~5, ~144/77
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0485{{c}}, ~3/2 = 701.9606{{c}}, ~5/4 = 386.2167{{c}}, ~144/77 = 1084.3250{{c}}
+: [[error map]]: {{val| +0.049 +0.054 -0.000 -0.127 -0.129 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9739{{c}}, ~5/4 = 386.2460{{c}}, ~144/77 = 1084.2953{{c}}
+: error map: {{val| 0.000 +0.019 -0.068 -0.235 -0.256 }}
+{{Optimal ET sequence|legend=1| 19, 22, 31, 41, 53, 72, 166, 167, 176, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}
+[[Badness]] (Sintel): 0.777
+== Chrysic (43923/43904) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: [[43923/43904]]
+{{Mapping|legend=1| 1 0 0 3 4 | 0 1 0 3 2 | 0 0 1 0 0 | 0 0 0 -4 -3 }}
+: Mapping generators: ~2, ~3, ~5, ~33/14
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0167{{c}}, ~3/2 = 701.9323{{c}}, ~5/4 = 386.2803{{c}}, ~33/28 = 284.2370{{c}}
+: [[error map]]: {{val| +0.017 -0.006 -0.000 +0.056 -0.114 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9341{{c}}, ~5/4 = 386.2927{{c}}, ~33/28 = 284.2352{{c}}
+: error map: {{val| 0.000 -0.021 -0.021 +0.036 -0.155 }}
+{{Optimal ET sequence|legend=1| 17c, 26, 29, 34d, 43, 46, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 2715, 2985, 3327e, 3479, 3821, 4091e, 4433e, 5539e }}
+[[Badness]] (Sintel): 0.770
+== Elysic (117649/117612) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: [[117649/117612]]
+{{Mapping|legend=1| 1 0 0 0 -1 | 0 2 0 0 -5 | 0 0 1 0 0 | 0 0 0 1 3 }}
+: Mapping generators: ~2, ~343/198, ~5, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0027{{c}}, ~343/198 = 950.9861{{c}}, ~5/4 = 386.3082{{c}}, ~7/4 = 968.7558{{c}}
+: [[error map]]: {{val| +0.003 +0.017 -0.000 -0.065 +0.033 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/198 = 950.9850{{c}}, ~5/4 = 386.3105{{c}}, ~7/4 = 968.7571{{c}}
+: error map: {{val| 0.000 +0.015 -0.003 -0.069 +0.028 }}
+{{Optimal ET sequence|legend=1| 49, 58, 72, 130, 193, 198, 212, 270, 342, 612, 954, 1084, 1354, 1547, 1817, 2159, 3243, 3513, 5060, 5402, 5672, 8915d, 11074d }}
+[[Badness]] (Sintel): 0.901
 == Olympic (131072/130977) ==
+Olympic is highly accurate yet easy to conceptualize and notate on the staff. It equates [[33/32]] with ([[64/63]])<sup>2</sup>, and [[36/35]] with [[1053/1024]], reducing the four [[formal comma]]s required for the [[13-limit]] to two. Therefore it can be notated in [[chain-of-fifths notation]] with two additional set of accidentals, one for the septimal comma step, and the other for the [[aberschisma]] step. It can be viewed as the [[detemperament]] of [[cassaschismic]] that decouples the Pythagorean and septimal commas.
+It also has a [[strong extension]] to include [[29/1|prime 29]] though with higher complexity.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 861: / Line 987: @@
 : Mapping generators: ~2, ~3, ~5, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.103{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 969.012{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9460{{c}}, ~3/2 = 702.0489{{c}}, ~5/4 = 386.4217{{c}}, ~7/4 = 968.9839{{c}}
+: [[error map]]: {{val| -0.054 +0.040 -0.000 +0.050 +0.038 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0853{{c}}, ~5/4 = 386.4125{{c}}, ~7/4 = 969.0242{{c}}
+: error map: {{val| 0.000 +0.130 +0.099 +0.198 +0.207 }}
 {{Optimal ET sequence|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}
+[[Badness]] (Sintel): 2.05
 === Tridecimal olympic ===
+==== 13-limit ====
 [[Subgroup]]: 2.3.5.7.11.13
-[[Comma list]]: 2080/2079, 4096/4095
+[[Comma list]]: [[2080/2079]], [[4096/4095]]
 {{Mapping|legend=1| 1 0 0 0 17 12 | 0 1 0 0 -5 -2 | 0 0 1 0 0 -1 | 0 0 0 1 -2 -1 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.103{{c}}, ~5/4 = 386.298{{c}}, ~7/4 = 969.007{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9638{{c}}, ~3/2 = 702.0645{{c}}, ~5/4 = 386.3208{{c}}, ~7/4 = 968.9712{{c}}
+: [[error map]]: {{val| -0.036 +0.073 -0.065 +0.073 +0.128 -0.166 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0884{{c}}, ~5/4 = 386.3269{{c}}, ~7/4 = 969.0015{{c}}
+: error map: {{val| 0.000 +0.133 +0.013 +0.176 +0.237 -0.033 }}
 {{Optimal ET sequence|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }} *
 <nowiki>*</nowiki> [[optimal patent val]]: [[3044edo|3044]]
+[[Badness]] (Sintel): 0.292
+===== 2.3.5.7.11.13.29 subgroup =====
+Subgroup: 2.3.5.7.11.13.29
+Comma list: 2080/2079, 3510/3509, 4096/4095
+Mapping: {{mapping| 1 0 0 0 17 12 -21 | 0 1 0 0 -5 -2 11 | 0 0 1 0 0 -1 0 | 0 0 0 1 -2 -1 3 }}
+Optimal tunings:
+* WE: ~2 = 1199.960{{c}}, ~3/2 = 702.008{{c}}, ~5/4 = 386.247{{c}}, ~7/4 = 968.895{{c}}
+: error map: {{val| -0.040 +0.053 -0.067 +0.069 +0.164 -0.171 +0.048 }}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.070{{c}}, ~5/4 = 386.337{{c}}, ~7/4 = 969.013{{c}}
+: error map: {{val| 0.000 +0.115 +0.023 +0.187 +0.304 -0.018 +0.237 }}
+{{Optimal ET sequence|legend=0| 41, 53j, 87, 130j, 176, 183, 217, 224, 270, 311, 487, 494, 1205, 1699 }}
+Badness (Sintel): 0.505
+==== Commas 2080/2079, 3250/3249, 4096/4095 ====
+This is a [[weak extension]] of olympic, requiring the split of the prime 7 in half; or alternatively, the septimal comma or the garischisma.
+Subgroup: 2.3.5.7.11.13.19
+Comma list: 2080/2079, 3250/3249, 4096/4095
+Mapping: {{mapping| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}
+Optimal tunings:
+* WE: ~2 = 1199.9639{{c}}, ~3/2 = 702.0643{{c}}, ~5/4 = 386.3218{{c}}, ~247/132 = 1084.4676{{c}}
+: error map: {{val| -0.036 +0.073 -0.064 +0.073 +0.129 -0.166 -0.004 }}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0887{{c}}, ~5/4 = 386.3257{{c}}, ~247/132 = 1084.5006{{c}}
+: error map: {{val| 0.000 +0.134 +0.012 +0.175 +0.236 -0.032 +0.135 }}
+{{Optimal ET sequence|legend=0| 41, 53, 84, 94, 137, 176, 217, 270, 581, 851, 935, 1205, 2193, 2463, 2774e }}
+Badness (Sintel): 0.618
+===== 2.3.5.7.11.13.19.29 subgroup =====
+Subgroup: 2.3.5.7.11.13.19.29
+Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095
+Mapping: {{mapping| 1 0 0 1 15 11 -13 6 -18 | 0 1 0 0 -5 -2 5 -2 -11 | 0 0 1 0 0 -1 2 1 0 | 0 0 0 2 -4 -2 5 -1 6 }}
+Optimal tunings:
+* WE: ~2 = 1199.9594{{c}}, ~3/2 = 702.0487{{c}}, ~5/4 = 386.3262{{c}}, ~247/132 = 1084.4677{{c}}
+: error map: {{val| -0.041 +0.053 -0.069 +0.069 +0.163 -0.170 +0.005 +0.048 }}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0709{{c}}, ~5/4 = 386.3327{{c}}, ~247/132 = 1084.5064{{c}}
+: error map: {{val| -0.000 +0.116 +0.019 +0.187 +0.302 -0.015 +0.171 +0.241 }}
+{{Optimal ET sequence|legend=0| 41, 53j, 94, 137j, 176, 217, 270, 487, 581j, 718, 894, 935, 988, 1205, 2193, 3398dehj}}
+Badness (Sintel): 1.10
+==== Metaolympic ====
+Metaolympic is the above extension with prime 17 added. It is better without prime 17, but prime 17 can be added easily into the temperament by also tempering out 4200/4199, which also takes advantage of the split prime 7 at the cost of higher complexity.
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 2080/2079, 4096/4095, 35000/34969
+Mapping: {{mapping| 1 0 0 1 15 11 -13 | 0 1 0 0 -5 -2 5 | 0 0 1 0 0 -1 2 | 0 0 0 2 -4 -2 5 }}
+: mapping generators: ~2, ~3, ~5, ~187/100
+Optimal tunings:
+* WE: ~2 = 1199.9530{{c}}, ~3/2 = 702.0493{{c}}, ~5/4 = 386.2541{{c}}, ~187/100 = 1084.4393{{c}}
+: error map: {{val| -0.047 +0.047 -0.154 +0.006 +0.208 -0.088 +0.184 }}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0798{{c}}, ~5/4 = 386.2658{{c}}, ~187/100 = 1084.4811{{c}}
+: error map: {{val| 0.000 +0.125 -0.057 +0.136 +0.359 +0.094 +0.363 }}
+{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2815eg }}
+Badness (Sintel): 1.59
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199
+Mapping: {{mapping| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}
+Optimal tunings:
+* WE: ~2 = 1199.9539{{c}}, ~3/2 = 702.0464{{c}}, ~5/4 = 386.2644{{c}}, ~187/100 = 1084.4405{{c}}
+: error map: {{val| -0.046 +0.045 -0.141 +0.009 +0.227 -0.088 0.192 -0.058 }}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0768{{c}}, ~5/4 = 386.2655{{c}}, ~187/100 = 1084.4813{{c}}
+: error map: {{val| 0.000 +0.122 -0.048 +0.137 +0.373 +0.090 +0.367 +0.118 }}
+{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2451eg }}
+Badness (Sintel): 0.764
+===== 2.3.5.7.11.13.17.19.29 subgroup =====
+Subgroup: 2.3.5.7.11.13.17.19.29
+Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095, 4200/4199
+Mapping: {{mapping| 1 0 0 1 15 11 -13 6 -18 | 0 1 0 0 -5 -2 5 -2 -11 | 0 0 1 0 0 -1 2 1 0 | 0 0 0 2 -4 -2 5 -1 6 }}
+Optimal tunings:
+* WE: ~2 = 1199.9561{{c}}, ~3/2 = 702.0552{{c}}, ~5/4 = 386.2577{{c}}, ~187/100 = 1084.4387{{c}}
+: error map: {{val| -0.044 +0.056 -0.144 +0.008 +0.212 -0.080 +0.205 -0.068 -0.031 }}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0795{{c}}, ~5/4 = 386.2631{{c}}, ~187/100 = 1084.4800{{c}}
+: error map: {{val| 0.000 +0.124 -0.051 +0.134 +0.365 +0.090 +0.368 +0.111 +0.177 }}
+{{Optimal ET sequence|legend=0| 53j, 94, 137j, 176g, 217, 270, 311, 487, 581j, 718, 935, 1205g, 1516e, 2234eg, 2721egg }}
+Badness (Sintel): 1.27
+== Alphaxenic (131769/131072) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 131769/131072
+{{Mapping|legend=1| 2 1 0 0 8 | 0 2 0 0 -1 | 0 0 1 0 0 | 0 0 0 1 0 }}
+: mapping generators: ~363/256, ~16/11, ~5, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}, ~5/4 = 385.6773{{c}}, ~7/4 = 968.1894{{c}}
+: [[error map]]: {{val| +0.318 -0.094 -0.000 -0.000 -0.897 }}
+* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}, ~5/4 = 385.7900{{c}}, ~7/4 = 968.1927{{c}}
+: error map: {{val| 0.000 -0.491 -0.524 -0.633 -2.050 }}
+{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}
+[[Badness]] (Sintel): 23.0
 == Seascape (160083/160000) ==
 [[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: [[Seascape comma|160083/160000]]
+[[Comma list]]: [[160083/160000]]
 {{Mapping|legend=1| 1 0 0 0 4 | 0 2 0 0 -3 | 0 0 1 0 2 | 0 0 0 1 -1 }}
 : Mapping generators: ~2, ~400/231, ~5, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~400/231 = 950.960{{c}}, ~5/4 = 386.417{{c}}, ~7/4 = 968.751{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0285{{c}}, ~400/231 = 950.9641{{c}}, ~5/4 = 386.3335{{c}}, ~7/4 = 968.7129{{c}}
+: [[error map]]: {{val| +0.028 -0.027 +0.077 -0.056 -0.085 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 950.9492{{c}}, ~5/4 = 386.3638{{c}}, ~7/4 = 969.7159{{c}}
+: error map: {{val| 0.000 -0.057 +0.050 -0.110 -0.154 }}
+{{Optimal ET sequence|legend=1| 14c, 15, 19, 29, 39d, 43, 53, 58, 72, 130, 183, 198, 212, 255, 270, 342, 612, 954, 1084, 1354, 1696, 2833e, 3575, 3917e, 4187e, 4529de, 5883de }}
-{{Optimal ET sequence|legend=1| 15, 19, 29, 43, 53, 58, 68, 72 }}
+[[Badness]] (Sintel): 0.596
 == Fifthchromismic (1610510/1594323) ==
@@ Line 895: / Line 1,165: @@
 [[Comma list]]: [[1610510/1594323]]
-{{Mapping|legend=1| 1 0 4 0 -1 | 0 1 3 0 2 | 0 0 -5 0 1 | 0 0 0 1 0 }}
+{{Mapping|legend=1| 1 0 -6 0 1 | 0 1 3 0 2 | 0 0 5 0 -1 | 0 0 0 1 0 }}
-: Mapping generators: ~2, ~3, ~22/9, ~7
+: Mapping generators: ~2, ~3, ~18/11, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.738{{c}}, ~11/9 = 344.406{{c}}, ~7/4 = 968.826{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9757{{c}}, ~3/2 = 702.7610{{c}}, ~18/11 = 855.5656{{c}}, ~7/4 = 968.8735{{c}}
+: [[error map]]: {{val| -0.024 +0.782 -0.130 -0.001 -1.434 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.7565{{c}}, ~18/11 = 855.5885{{c}}, ~7/4 = 968.8591{{c}}
+: error map: {{val| 0.000 +0.801 -0.102 +0.033 -1.393 }}
-{{Optimal ET sequence|legend=1| 77, 80, 87, 94, 167 }}
+{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 251, 258e, 338, 432, 519 }}
+[[Badness]] (Sintel): 97.2
 === Fifthchroma ===
-{{see also| Fifth-chroma temperaments }}
+{{See also| Fifth-chroma temperaments }}
 [[Subgroup]]: 2.3.5.7.11.13
@@ Line 909: / Line 1,185: @@
 [[Comma list]]: 352/351, 951665/944784
-{{Mapping|legend=1| 1 0 4 0 -1 4 | 0 1 3 0 2 -1 | 0 0 -5 0 1 1 | 0 0 0 1 0 0 }}
+{{Mapping|legend=1| 1 0 -6 0 1 6 | 0 1 3 0 2 -1 | 0 0 5 0 -1 -1 | 0 0 0 1 0 0 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8431{{c}}, ~3/2 = 702.9005{{c}}, ~18/11 = 855.4090{{c}}, ~7/4 = 969.1386{{c}}
+: [[error map]]: {{val| -0.157 +0.789 -0.097 -0.001 -1.396 +0.379 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.9326{{c}}, ~18/11 = 855.5400{{c}}, ~7/4 = 969.1437{{c}}
+: error map: {{val| 0.000 +0.978 +0.184 +0.318 -0.993 +1.000 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.776{{c}}, ~11/9 = 344.410{{c}}, ~7/4 = 968.826{{c}}
+{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 181, 244, 251, 261, 338, 599b }}
-{{Optimal ET sequence|legend=1| 77, 80, 87, 94, 167 }}
+[[Badness]] (Sintel): 7.63
 ==== 17-limit ====
@@ Line 920: / Line 1,202: @@
 Comma list: 352/351, 715/714, 155771/155520
-Mapping: {{mapping| 1 0 4 0 -1 4 6 | 0 1 3 0 2 -1 3 | 0 0 -5 0 1 1 -3 | 0 0 0 1 0 0 -1 }}
+Mapping: {{mapping| 1 0 -6 0 1 6 0 | 0 1 3 0 2 -1 3 | 0 0 5 0 -1 -1 3 | 0 0 0 1 0 0 -1 }}
+Optimal tunings:
+* WE: ~2 = 1199.8126{{c}}, ~3/2 = 702.8981{{c}}, ~11/9 = 855.3764{{c}}, ~7/4 = 969.3543{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9376{{c}}, ~11/9 = 855.5306{{c}}, ~7/4 = 969.4313{{c}}
-Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~3/2 = 702.713{{c}}, ~11/9 = 344.403{{c}}, ~7/4 = 969.193{{c}}
+{{Optimal ET sequence|legend=0| 70, 73g, 77, 80, 87, 94, 157, 167, 181, 251, 261, 338, 432, 599bg }}
-{{Optimal ET sequence|legend=0| 77, 80, 87, 94, 167 }}
+Badness (Sintel): 3.65
 ==== 19-limit ====
@@ Line 931: / Line 1,217: @@
 Comma list: 352/351, 715/714, 1463/1458, 2057/2052
-Mapping: {{mapping| 1 0 4 0 -1 4 6 2 | 0 1 3 0 2 -1 3 4 | 0 0 -5 0 1 1 -3 -1 | 0 0 0 1 0 0 -1 -1 }}
+Mapping: {{mapping| 1 0 -6 0 1 6 0 0 | 0 1 3 0 2 -1 3 4 | 0 0 5 0 -1 -1 3 1 | 0 0 0 1 0 0 -1 -1 }}
+Optimal tunings:
+* WE: ~2 = 1199.8128{{c}}, ~3/2 = 702.9226{{c}}, ~11/9 = 855.3621{{c}}, ~7/4 = 969.3187{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9618{{c}}, ~11/9 = 855.5163{{c}}, ~7/4 = 969.3961{{c}}
+{{Optimal ET sequence|legend=0| 70, 73gh, 77, 80, 87, 94, 157, 167h, 181, 251, 261, 338, 599bg }}
+Badness (Sintel): 2.63
+== Quartismic (117440512/117406179) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 117440512/117406179
+{{Mapping|legend=1| 1 0 0 1 5 | 0 1 0 1 -1 | 0 0 1 0 0 | 0 0 0 5 1 }}
+: mapping generators: ~2, ~3, ~5, ~33/32
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9875{{c}}, ~3/2 = 701.9753{{c}}, ~5/4 = 386.3387{{c}}, ~33/32 = 53.3743{{c}}
+: [[error map]]: {{val| -0.012 +0.008 -0.000 -0.004 +0.031 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9802{{c}}, ~5/4 = 386.3343{{c}}, ~33/32 = 53.3730{{c}}
+: error map: {{val| 0.000 +0.025 +0.021 +0.019 +0.075 }}
+{{Optimal ET sequence|legend=1| 21, 22, 43, 46, 65d, 68, 89, 111, 159, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2159, 2653, 3125, 3395, 7060, 7554, 10949e, 14614e, 15850ee, 22168bdee, 23404bcdee, 26799bcdeee, 34353bcdeeee }}
+[[Badness]] (Sintel): 5.00
+=== Tridecimal quartismic ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 6656/6655, 123201/123200
+{{Mapping|legend=1| 1 0 0 1 5 6 | 0 1 0 1 -1 -3 | 0 0 1 0 0 1 | 0 0 0 5 1 3 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9884{{c}}, ~3/2 = 701.9674{{c}}, ~5/4 = 386.3432{{c}}, ~33/32 = 53.3763{{c}}
+: [[error map]]: {{val| -0.012 +0.001 +0.006 -0.000 +0.045 -0.016 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9728{{c}}, ~5/4 = 386.3385{{c}}, ~33/32 = 53.3748{{c}}
+: error map: {{val| 0.000 +0.018 +0.025 +0.021 +0.084 +0.017 }}
-Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~3/2 = 702.736{{c}}, ~11/9 = 344.417{{c}}, ~7/4 = 969.159{{c}}
+{{Optimal ET sequence|legend=1| 22, 43f, 46, 65d, 89f, 111, 159, 224, 270, 494, 764, 1012, 1236, 1506, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816dee }}
-{{Optimal ET sequence|legend=0| 77, 80, 87, 94, 167h }}
+[[Badness]] (Sintel): 0.456
-[[Category:Temperaments]]
-[[Category:Lists of temperaments]]
 [[Category:Temperament collections]]
-[[Category:Regular temperament theory]]
+[[Category:Catalogs of rank-4 temperaments]]
-[[Category:Rank 4| ]] <!-- main article -->